Algebra Of Real Rank Research Articles

Additive mappings on C ∗ -algebras sub-preserving absolute values of products

Let A be a C * -algebra of real rank zero and B be a C * -algebra with unit I.I t is shown that if φ : A -→ B is an additive mapping which satisfies |φ(A)φ(B) |≤ φ(|AB| )f or every A, B ∈ A+ and φ(A )= I for some A ∈ As withA �≤ 1, then the restriction of mapping φ to As is a Jordan homomorphism, where As denotes the set of all self-adjoint elements. We will also show that if φ is surjective preserving the product and an absolute value, then φ is a C-linear or C-antilinear *-homomorphism on A. MSC: Primary 47B49; Secondary 46L05; 47L30

Lifting Defects for Nonstable K0-theory of Exchange Rings and C*-algebras

The assignment (nonstable K0-theory), that to a ring R associates the monoid V( R ) of Murray-von Neumann equivalence classes of idempotent infinite matrices with only finitely nonzero entries over R, extends naturally to a functor. We prove the following lifting properties of that functor: By using categorical tools (larders, lifters, CLL) from a recent book from the author with P. Gillibert, we deduce that there exists a unital exchange ring of cardinality $\aleph_3$ (resp., an $\aleph_3$ -separable unital C*-algebra of real rank 1) R, with stable rank 1 and index of nilpotence 2, such that V( R ) is the positive cone of a dimension group but it is not isomorphic to V( B ) for any ring B which is either a C*-algebra of real rank 0 or a regular ring.

Linear Maps on C*-Algebras Preserving the Set of Operators that are Invertible in

AbstractFor C*-algebras of real rank zero, we describe linear maps ϕ on that are surjective up to ideals , and π(A) is invertible in if and only if π(ϕ(A)) is invertible in , where A ∈ and π : → is the quotient map. We also consider similar linear maps preserving zero products on the Calkin algebra.

Linear maps preserving Fredholm and Atkinson elements of C *-algebras

Let A be a unital C*-algebra of real rank zero and B be a unital semisimple complex Banach algebra. We characterize linear maps from A onto B preserving different essential spectral sets and quantities such as the essential spectrum, the (left, right) essential spectrum, the Weyl spectrum, the index and the essential spectral radius.

C* -Algebras of Real Rank Zero Whose K0's are not Riesz Groups

AbstractExamples are constructed of stably finite, imitai, separable C* -algebras A of real rank zero such that the partially ordered abelian groups K0(A) do not satisfy the Riesz decomposition property. This contrasts with the result of Zhang that projections in C* -algebras of real rank zero satisfy Riesz decomposition. The construction method also produces a stably finite, unital, separable C* -algebra of real rank zero which has the same K-theory as an approximately finite dimensional C*-algebra, but is not itself approximately finite dimensional.

C*-ALGEBRAS WITH REAL RANK ZERO AND THEIR CORONA AND MULTIPLIER ALGEBRAS PART IV

By proving various equivalent versions of the generalized Weyl-von Neumann theorem, we investigate the structure of projections in the multiplier algebra [Formula: see text] of certain C*-algebra [Formula: see text] with real rank zero. For example, we prove that [Formula: see text] if and only if any two projections in [Formula: see text] are simultaneously quasidiagonal. In case [Formula: see text] is a purely infinite simple C*-algebra, [Formula: see text] if and only if any two projections in [Formula: see text] are simultaneously quasidiagonal. If [Formula: see text] is one of the Cuntz algebras, or one of finite factors or type III factors, then any two projections in [Formula: see text] are simultaneously quasidiagonal. On the other hand, if [Formula: see text] is one of the Bunce-Deddens algebras or one of the irrational rotation algebras of real rank zero, then there exist two projections in [Formula: see text] which are not simultaneously quasidiagonal.